Quantitative cooling histories from stranded diffusion profiles

Abstract

Stranded elemental or isotopic diffusion profiles in geological materials have the potential to reveal information on the thermal history of the host sample. In the specific case of a concentration step that is established at high temperature, the extent of diffusive relaxation during cooling depends on the details of the cooling path and the Arrhenius diffusion law of the species of interest: In principle, a measured profile in a sample can provide quantitative information on the nature of the cooling path if the diffusion law is known. Using a combination of mathematics and numerical simulations, we derive a simple relationship describing the extent of profile relaxation (as gauged by the slope S ₀ of a diffusion profile) as a function of the initial temperature (T _i) and cooling rate ($\dot{T}$) of the system and the activation energy (E _a) and pre-exponential factor (D ₀) for diffusion:

$$\log S_{0} = 2.504 - \frac{1}{2}\log D_{0} - \log T_{\text{i}} + \frac{1}{2}\log E_{\text{a}} + \frac{1}{2}\log \dot{T} + \left( {26.11\frac{{E_{\text{a}} }}{{T_{\text{i}} }}} \right)$$

The initial temperature T _i is expressed in K, $\dot{T}$ is in °/s, D ₀ is in m²/s, and E _a is in kJ/mol. The slope of the profile of interest can be estimated either at the midpoint of an interdiffusion profile or at a crystal margin. In the former case, concentrations are normalized to a difference of 100 between the upper (=100) and lower (=0) initial concentration plateaus. For profiles at crystal margins, the normalization range is 0 to 50. The equation above applies equally well to linear and exponential cooling paths because the extent of relaxation indicated by S ₀ is essentially the same for a given linear cooling path and an exponential one characterized by the same initial cooling rate. Cooling from the top of parabolic T–t “dome” results in more extensive profile relaxation; this is also well described by the above equation if the leading constant 2.504 is changed to 2.165. If S ₀ of a stranded profile has been characterized in the laboratory, and if the Arrhenius law of the diffusant is known, the above equation can be solved uniquely for one of the cooling path parameters (T _i or $\dot{T}$) if the other—which will usually be T _i—is constrained by phase equilibria or the geological context of the sample. Alternatively, if a sample exhibits stranded profiles for two diffusants having different E _a and D ₀ values, two versions of the above equation can be solved simultaneously for both the initial temperature and the cooling rate. The equation above can be implemented for purposes other than estimating T–t histories: e.g., assessing whether an observed concentration profile is truly the result of diffusion or a consequence of changing phase composition during growth. Our approach also raises the possibility not only of cross-checking multiple laboratory-based diffusion laws but also of estimating Arrhenius parameters for uncharacterized diffusants.